*which*one it is: usually, it's the finite ordinals $\omega$.) Consequently, if you eliminate the carrier set, then everything else goes with it.

The approach described in the previous post starts with a model (structured set) $\mathcal{A}$, and then identifies the abstract structure with a kind of

*ramsified proposition*that categorically axiomatizes $\mathcal{A}$.

There is another approach. I'd read about this several years ago, but I'd forgotten about it. It's category-theoretic and it was described to me by one of our MCMP graduate students, Hans-Christoph Kotzsch, a couple of weeks ago. On this view,

*abstract structures are objects in categories*. The objects in a category $C$ needn't be regarded as built-up from a carrier set. And one can talk about something akin to "elements" of an object $X$ in a category $C$ by identifying such elements with morphisms $1 \rightarrow X$, where $1$ is a terminal object of $C$.

The idea is developed in a 2011

*Synthese*article, "Category-Theoretic Structure and Radical Ontic Structural Realism" by Jonathan Bain (and in these slides too).

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